L11n351

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,15,4,14 X_11,20,12,21 X_7,18,8,19 X_9,13,10,22 X_21,17,22,16 X_17,8,18,9 X_15,11,16,10 X_19,12,20,5 X₂₅₃₆ X_13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 7, -5, 8, -3, 9}, {-11, 2, -8, 6, -7, 4, -9, 3, -6, 5}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-uv^{3}w+uv^{3}+uv^{2}w^{3}-3uv^{2}w^{2}+4uv^{2}w-2uv^{2}-uvw^{3}+4uvw^{2}-3uvw+uv-v^{2}w^{3}+3v^{2}w^{2}-4v^{2}w+v^{2}+2vw^{3}-4vw^{2}+3vw-v-w^{3}+w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+12q^{2}-13q+15-13q^{-1}+11q^{-2}-6q^{-3}+3q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-4}+a^{4}z^{-2}+2a^{4}-a^{-4}+a^{2}z^{4}+2z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}-5a^{2}+2a^{-2}-z^{6}-2z^{4}-z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+5a^{2}z^{8}+5z^{8}a^{-2}+10z^{8}+3a^{3}z^{7}+5az^{7}+7z^{7}a^{-1}+5z^{7}a^{-3}-12a^{2}z^{6}-5z^{6}a^{-2}+3z^{6}a^{-4}-20z^{6}-3a^{3}z^{5}-13az^{5}-18z^{5}a^{-1}-7z^{5}a^{-3}+z^{5}a^{-5}+6a^{4}z^{4}+25a^{2}z^{4}-5z^{4}a^{-4}+24z^{4}+4a^{3}z^{3}+12az^{3}+12z^{3}a^{-1}+2z^{3}a^{-3}-2z^{3}a^{-5}-11a^{4}z^{2}-29a^{2}z^{2}-z^{2}a^{-2}+3z^{2}a^{-4}-22z^{2}-5a^{3}z-8az-3za^{-1}+za^{-3}+za^{-5}+6a^{4}+13a^{2}-a^{-4}+9+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

5

1

-4

5

7

2

5

3

6

5

-1

1

9

7

2

-1

7

9

2

-3

4

6

-2

-5

2

7

5

-7

1

4

-3

-9

3

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-4$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$